A Resolution-Based Calculus for Preferential Logics

Abstract

The vast majority of modal theorem provers implement modal tableau, or backwards proof search in (cut-free) sequent calculi. The design of suitable calculi is highly non-trivial, and employs nested sequents, labelled sequents and/or specifically designated transitional formulae. Theorem provers for first-order logic, on the other hand, are by and large based on resolution. In this paper, we present a resolution system for preference-based modal logics, specifically Burgess’ system Open image in new window. Our main technical results are soundness and completeness. Conceptually, we argue that resolution-based systems are not more difficult to design than cut-free sequent calculi but their purely syntactic nature makes them much better suited for implementation in automated reasoning systems.

Supplementary material

Proofs

The next two proofs were automatically generated by a prototype prover which implements the calculus given in this paper. Only clauses needed in the refutation are shown. Also the inference rule Open image in new window is always applied together with Open image in new window, so clauses are already in simplified form. First, as part of the proof of Lemma 7, we show that for \(\varphi ,\lnot \psi \in b\), we have that \(\mathsf {Prefer}(a,b,B) \) and \(\varphi \Rightarrow \psi \in (a,A)\) is contradictory.

The following refutation is part of the proof of Lemma 8, where we show that, for \(\varphi , \psi \in b\), we have that \(\lnot ((b \vee \bigvee B) \Rightarrow \bigvee B)\) and \(\lnot (\varphi \Rightarrow \psi ) \in a\) is not Open image in new window-consistent